Let k be an algebraically closed field of characteristic 0. How to prove that $M_2$(k), the k-algebra of 2 by 2 matrices over k, is not isomorphic to the group ring of any finite group G over k.
The group ring of a finite group is simple iff the group is trivial.
Indeed, there is always a $k$-algebra morphism $kG\to k$, coming from the trivial representation which, if $kG$ is simple, must be injective: this is only possible if $\dim kG=1$, that is, if $G$ is trivial.
Now matrix algebras are simple, so you get what you want.